Janusz Pawlikowski

Decomposing Baire functions

We discuss in the paper the following problem: Given a function in a given Baire class, into “how many” (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

On Ideals of Subsets of the Plane and on Cohen Reals

Let be any proper ideal of subsets of the real line R which contains all finite subsets of R . We define an ideal * ∣ as follows: X ∈ * ∣ if there exists a Borel set B ⊂ R × R such that X ⊂ B and for any x ∈ R we have { y ∈ R : 〈 x, y 〉 ∈ B } ∈ . We show that there exists a family ⊂ * ∣ of power ω 1 such that ⋃ ∉ * ∣ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.

A characterization of strong measure zero sets

We show that a setXυR has strong measure zero iff for every closed measure zero setFυR,F+X has measure zero.

CROWDED AND SELECTIVE ULTRAFILTERS UNDER THE COVERING PROPERTY AXIOM

Abstract In the paper we formulate an axiom , which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to w 1, while 𝔠 = w 2: the independence number 𝔦, the reaping number 𝔯, the almost disjoint number 𝔞, and the ultrafilter base number 𝔲. We will also show that implies the existence of crowded and selective ultrafilters as well as nonselective P-points. In addition we prove that under every selective ultrafilter is w 1-generated. The paper finishes with the proof that holds in the iterated perfect set model.

On dense subsets of the measure algebra

We show that the minimal cardinality of a dense subset of the measure algebra is the same as the minimal cardinality of a base of the ideal of Lebesgue measure zero subsets of the real line.

Decomposing Borel functions and structure at finite levels of the Baire hierarchy

Abstract We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions of continuous functions with domains of type Π n 0 , for a fixed n ω . For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n = 1 and Semmes for n = 2 .

Every Sierpiński set is strongly meager

A set X of reals is strongly null (has strong measure zero, has property C), if, given any sequence of real numbers εn > 0, X can be covered by a sequence In of intervals with the length of In < εn (see [B], [S]). Borel [B] conjectured that every strongly null set is countable. Laver [L] showed Borel’s conjecture to be consistent with ZFC. On the other hand Sierpi ńsk [S] showed that the continuum hypothesis (CH) falsifies it. He proved that every Lusin set is strongly null. A Lusin set is an uncoutable set of reals which meets every meager set in a countable set. Lusin [Lu] (see also [M]) constructed such a set from CH. A counterpart of a Lusin set for the Lebesgue measure is a Sierpi ński set – an uncountable set of reals which meets every null set in a countable set. Sierpi ński [S1] showed that such sets exist under CH. A counterpart of a strongly null set is obtained from the Galvin-MycielskiSolovay [GMS] characterization of strongly null: X is strongly null iff for every meager setA there is a realx with X ∩ (A + x) empty (see [M] for a proof). Exchanging “meager” and “null” we get a definition of strongly meager: a set X of reals is strongly meager iff for every null set A there is a real x with X∩(A+x) empty. The measure analogue of Laver’s result was established by Carlson [C]: it is consistent with ZFC that every strongly meager set is countable. Galvin (see [M]) asked about the measure analogue of Sierpi ński’s theorem: Is every Sierpi ński set strongly meager? Thre are some partial results: Jasi ński and Weiss [JW] showed that if S is a Sierpínski set andA is a null Fσ set thenS ∩ (A + x) = ∅ for somex. Recl ́ aw [R1] generalized this and proved that if A is an Fσ subset of the plane with all horizontal sectionsAs (s ∈ S) null, then ⋃

Finite Support Iteration and Strong Measure Zero Sets

Any finite support iteration of posets with precalibre χ 1 which has the length of cofinality greater than ω 1 yields a model for the dual Borel conjecture in which the real line is covered by χ 1 strong measure zero sets

On sums of darboux and nowhere constant continuous functions

We show that the property (P) for every Darboux function g: R → R there exists a continuous nowhere constant function f: R → R such that f + g is Darboux follows from the following two propositions: (A) for every subset S of R of cardinality c there exists a uniformly continuous function f: R → [0, 1] such that f[S] = [0, 1], (B) for an arbitrary function h: R → R whose image h[R] contains a nontrivial interval there exists an A C R of cardinality c such that the restriction h | A of h to A is uniformly continuous, which hold in the iterated perfect set model.

Adding Dominating Reals With

Let B be the random real forcing. Miller [Mi] asked if there are ZFC models M ⊆ N such that forcing with B M over N adds a dominating real. A YES answer was provided by Judah and Shelah in [JS], where in a long and sophisticated construction they built such models. In this paper we prove that forcing with B V over V I , where I is the infinitely often equal real forcing of [Mi], adds a dominating real over V I . This greatly simplifies the YES answer to Millers question. Moreover it turns out that B may be replaced here by E , the eventually different real forcing of [Mi]. This answers the second part of Millers question. We also prove that both side by side products I × B and I × E add a Hechler dominating real over V . In this section we establish the main result of the paper; namely, we prove that forcing over V I with either of the posets B V or E V adds a dominating real over V I . First we recall the definitions of I and E from [Mi]. The infinitely often equal real forcing is the set ordered by extension. Miller [Mi] proves that I is ω ω bounding.